Theory Overview on Spectroscopy

arXiv:1108.2197v1 [hep-ph] 10 Aug 2011

Theory Overview on Spectroscopy

Ahmed Ali∗ Deutsches-Elektronen Synchrotron DESY, Notkestrasse 85, D-22607 Hamburg E-mail: [email protected]

A theoretical overview of the exotic spectroscopy in the charm and beauty quark sector is presented. These states are unexpected harvest from the e+ e− and hadron colliders and a permanent abode for the majority of them has yet to be found. We argue that some of these states, in particular the Yb (10890) and the recently discovered states Zb (10610) and Zb (10650), discovered by the Belle collaboration are excellent candidates for tetraquark states [bq][b¯ q], ¯ with q = u, d light quarks. Theoretical analyes of the Belle data carried out in the tetraquark context is reviewed.

The 13th International Conference on B-Physics at Hadron Machines - Beauty2011, April 04-08, 2011 Amsterdam, The Netherlands ∗ Speaker.

c Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence.

http://pos.sissa.it/

Spectroscopy-Overview

Ahmed Ali

1. Introduction The title of my talk is both ambitious and pretentious! I hasten to state that the mandate given to me is rather limited, namely to review the phenomenology of hadronic states discovered recently in the mass region of the charmonia and the bottomonia. Spearheaded by the experiments at the B factories and the Tevatron, with the experiments at the LHC as welcome new-comers, an impressive number of new states have been reported. Generically called X , Y and Z, these states defy a conventional quarkonia interpretation; this certainly holds for the majority of them. Their gross properties, such as the spin-parity assignments, masses, production mechanisms and decay modes, have been discussed in a number of comprehensive reviews [1, 2]. There have been a number of more recent developments in the field of quarkonium spectroscopy and I will confine myself just to their discussion. They involve the observation of the two charged bottomonium-like resonances by the Belle Collaboration [3] in the π ± ϒ(nS) (n = 1, 2, 3) and π ± hb (mP) (m = 1, 2) mass spectra that are produced in association with a single charged pion in e+ e− annihilation at energies near the ϒ(5S) resonance. Here hb (mP) are the P-wave spin-singlet bottomonia states. Calling the charged particles Zb (10610) and Zb (10650), their masses and the decay widths averaged over the five final states are, respectively, M[Zb (10610)] = 10608.4± 2.0 MeV, Γ[Zb (10610)] = 15.6 ± 2.5 MeV, and M[Zb (10650)] = 10653.2 ± 1.5 MeV, Γ[Zb (10650)] = 14.4 ± 3.2 MeV. The favoured quantum number assignments for both are I G (J P ) = 1+ (1+ ). This discovery was preceded by the observation of the hb (1P) and hb (2P) states, also by the Belle Collabora+1.03 ) tion [4] in the reaction e+ e− → hb (nP)π + π − , with the masses M[hb (1P)] = (9898.25 ± 1.06−1.07 +1.43 MeV and M[hb (1P)] = (10259.76 ± 0.64−1.03 ) MeV. These measurements yield hyperfine splitting in the bottomonium sector, defined as the mass difference between the P-wave spin-singlet state hb (mP) and the weighted average of the corresponding P-wave triplet states, χbJ (nP), ∆MHF (nP) ≡ +1.57 hM(n3 PJ )i − M(n1 P1 ), with ∆MHF (1P) = (1.62 ± 1.52) MeV and ∆MHF (2P) = (0.48−1.22 ) MeV. They are consistent with theoretical expectations and also with the hyperfine splitting measured in the charmonium sector ∆MHF = (0.14 ± 0.30) MeV [5], consistent with zero. Theoretically expected widths of hb (1P) and hb (2P) are of order 100 keV [6], which are too small to be measured by Belle. Still on the subject of hb (1P), the BaBar collaboration [7] has presented evidence of its production in the decay ϒ(3S) → π 0 hb (1P), followed by the decay hb (1P) → γηb (1S), in the distribution of the recoil mass against the π 0 at the mass M[hb (1P)] = (9902 ± 4 ± 1) MeV, which is consistent with the Belle measurements [4]. The width of hb (1P) is consistent with the experimental resolution, and the reported product branching ratio is B(ϒ(3S) → π 0 hb ) × B(hb → γηb ) = (3.7 ± 1.1 ± 0.7) × 10−4 . In this, and also in M[hb (1P)], the first error is statistical and the second systematic. The isospin-violating decay ϒ(3S) → π 0 hb (1P) is expected to have a branching fraction of about 10−3 [8, 9], and the branching fraction B(hb (1P) → γηb (1S)) ∼ (40−50)% [6]; hence, the measured product branching ratio is as anticipated theoretically. It is noteworthy that the decay ϒ(3S) → hb (1P)π + π − , which is suppressed by at least an order of magnitude compared to the decay ϒ(3S) → π 0 hb (1P) [8], has not been observed. The observation of the singlet P-state in the charmonium sector hc (1P) has also been reported this year by the CLEO collaboration [10] in the process e+ e− → π + π − hc (1P) at the center-of-mass energy Ec.m. = 4170 MeV. In fact, CLEO pioneered the technique of searching for peaks in the mass spectrum recoiling against the π 0 , and the 2


Ahmed Ali

resulting mass M[hc (1P)] = (3525.27 ± 0.17) MeV measured by this method is consistent with an earlier measurement of the hc (1P) mass from the decay ψ (2S) → π 0 hc [11]. The product branching ratio B(ψ (2S) → π 0 hc ) × B(hc → γηc ) = (4.19 ± 0.32 ± 0.45) × 10−4 is in agreement with theoretical expectations, and is also very similar to what has been reported by Babar for the corresponding hb (1P) product branching ratio, quoted above. However, there is an intriguing hint in the CLEO measurements of the cross section for e+ e− → hc (1P)π + π − , which rises at Ec.m. = 4260 MeV. Since this is close to the mass of the J PC = 1−− hadron Y (4260), which is a candidate for the hidden cc¯ tetraquark state, it would suggest that the mechanism e+ e− → Y (4260) → hc (1P)π + π − has something to do with the rise in the cross section. This remains to be confirmed in the next round of precise experiments.

2. Current experimental anomalies There is a number of anomalous features in the Belle data taken in the center-of-mass energy region near the ϒ(5S) mass. The first of these was reported some three years ago [12, 13] in the processes e+ e− → ϒ(1S)π + π − , ϒ(2S)π + π − , ϒ(3S)π + π − , measured in the center-of-mass energy range between 10.83 GeV and 11.02 GeV. The enigmatic features of the Belle data are (i) the anomalously large decay widths (or cross sections) for the mentioned final states, and (ii) the dipion invariant mass distributions recoiling against the ϒ(1S) and ϒ(2S) states, which are at variance with similar spectra measured in the transitions involving lower mass bottomonium states ϒ(nS) → ϒ(mS)π + π − (with m < n). To quantify the problem, the reported partial widths are Γ[ϒ(1S)π + π − )] = 0.59 ± 0.04 ± 0.09 MeV and Γ[ϒ(2S)π + π − )] = 0.85 ± 0.07 ± 0.16 MeV. Compared to the corresponding partial decay widths of the lower three ϒ(nS) (n = 2, 3, 4) states, Γ[ϒ(2S) → ϒ(1S)π + π − )] ∼ 6 keV, Γ[ϒ(3S) → ϒ(2S)π + π − )] ∼ 0.9 keV, and Γ[ϒ(4S) → ϒ(1S)π + π − )] ∼ 1.9 keV, the production of the ϒ(nS)π + π − in the energy region near the ϒ(5S) is larger by two to three orders of magnitude. The order keV partial widths are well-accounted for in the QCD multipole expansion [14, 15] based essentially on the Zweig-suppressed process shown in Fig. 1 (left-hand frame). The dipion invariant mass spectrum anticipated in the QCD multipole expansion is shown on the example of the decay ϒ(4S) → ϒ(1S)π + π − in Fig. 1 (right-hand frame) and compared with the data taken from the Belle collaboration at ϒ(4S) [16]. They are in excellent agreement with each other. Not so, for the dipionic transitions measured in the ϒ(5S) region, in which the dipionic mass spectra are dominated by the scalar meson f0 (980) and the tensor meson f2 (1270) (for the ϒ(1S)π + π − mode) and by the f0 (600) and f0 (980) mesons (for the ϒ(2S)π + π − mode). This is illustrated in Fig. 2 for the process e+ e− → ϒ(1S)π + π − which shows the distributions in the Mπ + π − (left-hand frame) and in the helicity angle (cos θ distribution (right-hand frame). The dipion mass spectrum measured near the ϒ(5S) clearly shows peaks at f0 (980) and f2 (1270). An interpretation of the process in terms of the production and decay of a J PC = 1−− tetraquark state [17, 18] (histograms and the solid curves) accounts well the experimental distributions. We will return to discuss the underlying dynamical model later in section 4 of this report. Not only are the cross sections for e+ e− → ϒ(nS)π + π − (n = 1, 2, 3) near the ϒ(5S) anomalously large by at least two orders of magnitude, the same holds for the production of the Pwave spin-singlet bottomonia states hb (mP) (m = 1, 2), for which the production cross sections for e+ e− → hb (1P)π + π − and e+ e− → hb (2P)π + π − are also anomalously large [4]. The ratios of the 3


Ahmed Ali

Figure 1: Left frame: Zweig-suppressed diagram for the transition ϒ(nS) → ϒ(mS)ππ with m < n, which forms the basis of the QCD estimates of the decay rates and distributions in heavy quarkonia dipionic transitions. Right frame: The dipion invariant mass spectrum Mππ measured in the decay ϒ(4S) → ϒ(1S)ππ by the Belle collaboration together with a theoretical curve based essentially on the diagram shown in the left frame. (From [16].) 1

2

(a)

dσfπ+π− /0.2

dσfπ+π− /(0.1 GeV)

2.5

0.8 0.6

1.5

0.4

1

0.5

0 0.2

(b)

0.2

0.4

0.6

0.8

1

1.2

0

1.4

-1

-0.5

Mπ+π− [GeV]

0

0.5

1

cos θ

Figure 2: Fit results of the Mπ + π − distribution (a) and the cos θ distribution (b) for e+ e− → Yb → ϒ(1S)π + π − , normalized by the measured cross section by Belle [12]. The histograms represent theoretical fit results based on the tetraquarks hypothesis, while the crosses are the Belle data. The solid curves in (a) show purely resonant contributions from the f0 (980) and f2 (1270). (From [18].)

production cross-sections in the indicated final states relative to that for the e+ e− → ϒ(2S)π + π − production are as follows [4]: +0.037 σ˜ [ϒ(1S)π + π − ] = 0.638 ± 0.065−0.056

σ˜ [ϒ(3S)π + π − ] = 0.517 ± 0.082 ± 0.070

+0.043 σ˜ [hb (1P)π + π − ] = 0.407 ± 0.07−0.076 +0.22 σ˜ [hb (2P)π + π − ] = 0.78 ± 0.09−0.10

(2.1)

We have already commented on the anomalous production cross sections in the ϒ(ns)π + π − modes near the ϒ(5S) region. The ratios given in the last two equations above for the hb (1P)π + π − and hb (2P)π + π − are found to be of order unity, a feature which violates theoretical expectations as the processes ϒ(5S) → hb (mP)π + π − involve heavy quark spin-flip, which are suppressed by 1/mb in the amplitude. It is obvious that the production mechanisms of all five processes involving ϒ(nS)π + π − (n = 1, 2, 3) and hb (mP)π + π − (m = 1, 2) are exotic. In particular, the true mechanisms at work avoid the Zweig-suppression seen in similar dipionic transitions and evade power suppression due to the spin-flip transitions for the hb (mP)π + π − case. It is worth recalling that no excess of the kind seen in the Belle measurements near the ϒ(5S) [12, 13, 4] is seen by them or any other experiment either at energies below or above the ϒ(5S) region. Any plausible theoretical explanation must account for all these features. 4


Ahmed Ali

These measurements have invoked a number of theoretical ideas. Particularly interesting is the suggestion by Bondar et al. [19], in which the resonances Zb (10610) and Zb (10650) are assumed mostly of a ’molecular’ type due to their respective proximity with the B∗ B¯ and B∗ B¯ ∗ thresholds. Thus, the internal dynamics of the states Zb (10610) and Zb (10650) is dominated by the coupling to meson pairs B∗ B¯ − BB¯ ∗ and B∗ B¯ ∗ , respectively. In particular, the bb¯ pair within the Zb (10610) and Zb (10650) is an equal mixture of a spin-triplet and spin-singlet with the relative phase orthogonal between the two resonances, i.e., 1 − − − |Zb (10610)i = √ 0− , ⊗ 1 − 1 ⊗ 0 ¯ ¯ ¯ Qq Qq bb¯ 2 bb 1 − − − . (2.2) |Zb (10650)i = √ 0− ⊗ 1 + 1 ⊗ 0 ¯ ¯ ¯ ¯ Qq Qq bb 2 bb

Here 0− and 1− stand for the para- and ortho-states with negative parity. The assignments (2.2) would predict that the mass difference M[Zb (10650)] − M[Zb (10610)] should be equal to that between the B and B∗ masses. The observed mass difference of 46 MeV [4] is in neat agreement with this argument. The spin-structure in (2.2) also suggests that the resonances Zb (10610) and Zb (10650) have the same decay width. This again is in agreement within measurement errors with the Belle data [4]: Γ[Zb (10610)] = 15.6 ± 2.5 MeV and Γ[Zb (10650)] = 14.4 ± 3.2 MeV. The maximal ortho-para mixing of the heavy quarks in the Zb (10610) and Zb (10650) resonances described by Eq. (2.2) also implies couplings of comparable strengths to channels with states of ortho- and para-bottomonium, leading to the following couplings of these resonances to the channels ϒ(nS)π ± and hb (mP)π ± [19]: Ch Eπ ~ϒ(nS) · (~Zb (10610) − ~Zb (10650)) ,

Cϒ (~pπ ×~hb ) · (~Zb (10610) + ~Zb (10650)) ,

(2.3)

where ~Zb (10610), ~Zb (10650) and ~hb denote the polarization vectors of the corresponding spin-1 states, and Eπ and ~pπ are the pion energy and its three-momentum, respectively; Ch and Cϒ are a priori unknown coupling constants to be determined by data. The amplitudes described by Eq. (2.3) applied to the decays ϒ(5S) → ϒ(nS)π + π − and ϒ(5S) → hb (mS)π + π − yield the right pattern of destructive and constructive interferences seen in the Dalitz distributions of these processes [4]. All of these arguments are plausible. Further variations on the molecular theme and predictions can be seen in [20, 21, 22, 23]. However, the structure suggested in Eq. (2.2) is a postulate not yet seen in decays other than those of the ϒ(5S). A particular case in point are the decays of the ϒ(6S), where the available phase space for the decays ϒ(6S) → ϒ(nS)π + π − and ϒ(6S) → hb (mP)π + π − are much larger. Hence, the implications of Eqs. (2.2) and (2.3) should be, at least qualitatively, very similar to those discussed in the context of the Belle data from the ϒ(5S) region. This remains to be tested. In addition, there are also some specific features of the Belle data which do not go hand-in-hand with the usual understanding of a hadronic molecule, the closest example of which is the Deuteron. The masses of the Zb (10610) and Zb (10650) are above the respective thresholds. The Deuteron mass, on the other hand, lies below the threshold by about 2.2 MeV. Also, the decay widths of the Zb (10610) and Zb (10650) are not particularly small, as one would expect for a hadron molecule. On the contrary, their decay widths are similar in order of magnitude as that of the ϒ(5S). This is also curious as the other ’hadronic molecule’ discussed at length in a similar context, namely the X (3872), has 5


Ahmed Ali

a much smaller (by at least an order of magnitude) decay width, with the current 90% C.L. limit being Γ[X (3872)] < 1.2 MeV [24]. In the rest of this writeup, I will take the point of view that all the five anomalous processes measured by Belle at energies near the ϒ(5S) mass [12, 13, 4] have very little to do with the ϒ(5S) decays. Following [17, 18, 25], I will argue here that the final states ϒ(nS)π + π − and hb (mP)π + π − are the decay products of the J PC = 1−− tetraquark Yb (10890), which lies in mass tantalizingly close to the ϒ(5S) mass. More precise experiments are needed to tell the two apart than is the case currently. In the context of the ϒ(nS)π + π − final states, this was suggested in [17, 18, 25] and the dynamical model was shown to be consistent with the observed cross sections. Also, the measured dipion invariant mass distributions show the predicted scalar-and tensor-meson resonant structure. Moreover, in the tetraquark context, it is easier to understand why the production cross sections for e+ e− → Yb (10890) → ϒ(nS)π + π − , which involves a 3 P → 3 S transition, and for e+ e− → Yb (10890) → hb (mS)π + π − , which involves a 3 P → 1 P transition, are comparable to each other. Detailed distributions, including the resonant Zb (10610) and Zb (10650) effects are still being worked out in the tetraquark picture.

3. Spectrum of bottom diquark-antidiquark states Much of the discussion of the tetraquark states involves the concept of diquarks (and antidiquarks) as effective degrees of freedom, which will be used here to calculate the mass spectra, production and decay of the tetraquark states. In particular, four-quark configurations in the tetraquarks are assumed not to play a dominant role. Following this, the mass spectrum of tetraquarks [bq][bq′ ] with q = u, d, s and c can be calculated using a Hamiltonian [26] (QQ)

H = 2mQ + HSS

¯ (QQ)

+ HSS

+ HSL + HLL ,

(3.1)

where: (QQ)

HSS

¯ (QQ) HSS

= 2(Kbq )3¯ [(Sb · Sq ) + (Sb¯ · Sq¯ )], = 2(Kbq¯ )(Sb · Sq¯ + Sb¯ · Sq ) + 2Kbb¯ (Sb · Sb¯ ) + 2Kqq¯ (Sq · Sq¯ ),

HSL = 2AQ (SQ · L + SQ · L), HLL = BQ

LQQ¯ (LQQ¯ + 1) . 2

(3.2)

¯ as the diquarks in the All diquarks, denoted here by Q are assumed to be in the color triplet (3), (6) representation do not show binding [27]. Here mQ is the constituent mass of the diquark [bq], (Kbq )3¯ is the spin-spin interaction between the quarks inside the diquarks, Kbq¯ are the couplings ranging outside the diquark shells, AQ is the spin-orbit coupling of diquark and BQ corresponds to the contribution of the total angular momentum of the diquark-antidiquark system to its mass. The overall factor of 2 is used customarily in the literature. As the isospin-breaking effects are estimated to be of order 5 - 8 MeV for the tetraquarks [bq][b¯ q] ¯ [25, 26], they are neglected in the mass estimates discussed below. 6


Ahmed Ali

The parameters involved in the above Hamiltonian (3.2) can be obtained from the known meson and baryon masses by resorting to the constituent quark model [29] H = ∑ mi + ∑ 2Ki j (Si · S j ), i

(3.3)

i< j

where the sum runs over the hadron constituents. The coefficient Ki j depends on the flavour of the constituents i, j and on the particular colour state of the pair. The constituent quark masses and the couplings Ki j for the colour singlet and anti-triplet states are given in [25]. To calculate the spin ¯ states explicitly, one uses the non-relativistic notation [28] SQ , SQ¯ ; J , spin interaction of the QQ where SQ and SQ¯ are the spin of diquark and antidiquark, respectively, and J is the total angular momentum. These states are then defined in terms of the direct product of the 2 × 2 matrices in spinor space, Γα , which can be written in terms of the Pauli matrices as:

σ2 1 (3.4) Γ0 = √ ; Γi = √ σ2 σi , 2 2 which then lead to the definition such as 0Q , 0Q¯ ; 0J = 21 (σ2 ) ⊗ (σ2 ). Others can be seen in [25]. The next step is the diagonalization of the Hamiltonian (3.1) using the basis of states with definite diquark and antidiquark spin and total angular momentum., There are two different pos sibilities [28]: Lowest lying [bq][b¯ q] ¯ states LQQ¯ = 0 and higher mass [bq][b¯ q] ¯ states LQQ¯ = 1 . The [bq][b¯ q] ¯ states LQQ¯ = 0 can be classified in terms of the six possible states involving the good (spin-0) and bad (spin-1) diquarks (here, P is the parity and C the charge conjugation) i. Two states with J PC = 0++ : ++ 0 = 0Q , 0Q¯ ; 0J ; ++′ 0 (3.5) = 1Q , 1 ¯ ; 0J . Q

ii. Three states with J = 1: ++ 1 1 = √ 0Q , 1Q¯ ; 1J + 1Q , 0Q¯ ; 1J ; 2 +− 1 1 = √ 0Q , 1Q¯ ; 1J − 1Q , 0Q¯ ; 1J ; +−′ 2 1 = 1Q , 1Q¯ ; 1J .

(3.6)

All these states have positive parity as both the good and bad diquarks have positive parity and LQQ¯ = 0. The difference is in the charge conjugation quantum number, the state |1++ i is even under charge conjugation, whereas |1+− i and |1+−′ i are odd. iii. One state with J PC = 2++ : ++ 2 (3.7) = 1Q , 1Q¯ ; 2J . Keeping in view that for LQQ¯ = 0 there is no spin-orbit and purely orbital term, the Hamiltonian (3.1) takes the form H = 2mQ + 2(Kbq )3¯ [(Sb · Sq ) + (Sb¯ · Sq¯ )] + 2Kqq¯ (Sq · Sq¯ ) +2(Kbq¯ )(Sb · Sq¯ + Sb¯ · Sq ) + 2Kbb¯ (Sb · Sb¯ ). 7

(3.8)


Ahmed Ali

The diagonalisation of the Hamiltonian (3.8) with the states defined above gives the eigenvalues which are needed to estimate the masses of these states. For the 1++ and 2++ states the Hamiltonian is diagonal with the eigenvalues [28] 1 1 M 1++ = 2mQ − (Kbq )3¯ + Kqq¯ − Kbq¯ + Kbb¯ , 2 2 1 1 ++ M 2 = 2m[bq] + (Kbq )3¯ + Kqq¯ + Kbq¯ + Kbb¯ . 2 2

(3.9) (3.10)

Mass of the constituent diquark can be estimated in one of two ways: We take the Belle data [12] as input and identify the Yb (10890) with the lightest of the 1−− states, Y[bq] , yielding a diquark mass m[bq] = 5.251 GeV. This procedure is analogous to what was done in [28], in which the mass of the diquark [cq] was fixed by using the mass of X (3872) as input, yielding m[cq] = 1.933 GeV. Instead, if we use this determination of m[cq] and use the formula m[bq] = m[cq] + (mb − mc ), which has the virtue that the mass difference mc − mb is well determined, we get m[bq] = 5.267 GeV, yielding a difference of 16 MeV. This can be taken as an estimate of the theoretical error on m[bq] , which then yields an uncertainty of about 30 MeV in the estimates of the tetraquark masses from this source alone. For the corresponding 0++ and 1+− tetraquark states, there are two states each, and hence the Hamiltonian is not diagonal. After diagonalising the 2 × 2 matrices, the masses of these states are obtained. We now discuss orbital excitations with LQQ¯ = 1 having both good and bad diquarks. Concentrating on the 1−− multiplet, we recall that there are eight tetraquark states [bq][b¯ q] ¯ (q = u, d), and the lightest isospin doublet is: (1) (3.11) MY[bq] SQ = 0, SQ¯ = 0, SQQ¯ = 0, LQQ¯ = 1 = m[bq] + λ1 + BQ ,

(2) and the next in mass is: MY[bq] SQ = 1, SQ¯ = 0, SQQ¯ = 1, LQQ¯ = 1 = 2m[bq] + ∆ + λ2 − 2AQ + BQ , and so on. Values of λi (i = 1, 2, 3), AQ and BQ are estimated in [25]. We identify the state Yb (10890) (1) with MY[bq] (in fact there are two of them, which differ in mass from each other by about 5 - 8 MeV, including isospin-breaking). This does not fix the quantity ∆, which is the mass difference of the good and the bad diquarks, i.e. ∆ = mQ (SQ = 1) − mQ (SQ = 0). Following Jaffe and Wilczek [27], the value of ∆ for diquark [bq] is estimated as ∆ = 202 MeV for q = u, d, s and c quarks. This is another source of potential uncertainty in estimating the tetraquark masses. The mass spectrum for the tetraquark states [bq][b¯ q] ¯ for q = u, d with J PC = 0++ , 1++ , 1+− , 1−− and 2++ states is plotted in Fig. 3 in the isospin-symmetry limit. It is difficult to quote a theoretical error on the masses shown, with ±50 MeV presumably a good guess. Other estimates of the tetraquark mass spectra in the charm and bottom quark sectors can be seen in [31, 32, 33]. 3.1 Estimates of the charged J P = 1+ tetraquark states In the tetraquark picture, one also anticipates a large number of charged states whose mass spectrum can be calculated in an analogous fashion as for their neutral counterparts just discussed. We would like to propose that the two charged J P = 1+ states Zb (10610) and Zb (10650) observed recently by the Belle Collaboration [3], and interpreted by them as the charged bottomonium states produced in the process ϒ(5S) → Zb± (10610) + π ∓ and ϒ(5S) → Zb± (10650) + π ∓ , are indeed ¯ for the positively charged state (its charged tetraquark states with the quark content Zb+ = [bu][b¯ d] 8


Ahmed Ali

charge conjugate being Zb− = [b¯ u][bd]). ¯ For the present discussion, they are produced in the decays PC −− of the J = 1 tetraquark Yb (10890). According to this interpretation, the decay chains involve Yb (10890) → ((Zb± (10610), Zb± (10650) + π ∓ → ϒ(ns)π + π − . A detailed dynamical model is under development with the aim of understanding the decay distributions in the kinematic variables available in these decays. We have estimated the masses of the isospin partners of Zb (10610) and Zb (10650), the two neutral J = 1 tetraquark states, denoted as |1+− i and |1+−′ i. The 2 × 2 non-diagonal mass matrix for the neutral J PC = 1+− states was, however, calculated numerically for ∆ = 0. If we ignore the isospin-breaking effects in the tetraquark masses, which are small, then the charged counterparts have the masses M[Zb (10610)] = 10.386 GeV and M[Zb (10650)] = 10.527 GeV, given in Fig. 3. As Zb (10610) involves one good and one bad diquark and Zb (10650) involves two bad diquarks, including the ∆-dependent term, the non-diagonal 2 × 2 mass matrix gets modified to the following form ! ∆ + κ κ 3 − ( − ) + − κ κ κ κ ¯ ¯ ¯ q q ¯ bq b q ¯ q q ¯ bb 3 bb 2 M(1+− ) = 2mQ + ∆ − + . (3.12) ∆ 2 2 + ( ) − κqq¯ − κbb¯ κ κbq¯ ¯ bq 3 2 p The two eigenvalues can be written as E = ± x2 + y2 , with x = ∆2 + (κbq )3¯ − κbq¯ and y = κqq¯ − κbb¯ , yielding r κqq¯ + κbb¯ ∆ 3 M[Zb (10650)] = 2mQ + ∆ − + ( + (κbq )3¯ − κbq¯ )2 + (κqq¯ − κbb¯ )2 , (3.13) 2 2 2 r κqq¯ + κbb¯ 3 ∆ M[Zb (10610)] = 2mQ + ∆ − − ( + (κbq )3¯ − κbq¯ )2 + (κqq¯ − κbb¯ )2 . (3.14) 2 2 2 Using the default values of the parameters [25] mQ = 5.251 GeV, (κqq¯ )0 = 318 MeV, (κbb¯ )0 = 36 MeV, (κbq¯ )0 = 23 MeV, (κbq )3 = 6 MeV (3.15) we have now the following predictions for the two charged tetraquark masses M[Zb (10610)] = 10.637 GeV; M[Zb (10650)] = 10.884 GeV, with ∆ = 202 MeV .

(3.16)

These estimates are to be compared with the masses of the J P = 1+ states Zb (10610) and Zb (10650) reported by the Belle Collaboration [3] M[Zb (10610)] = (10608 ± 2.0) MeV and M[Zb (10650)] = (10653.2 ± 1.5) MeV. They are in the right ball-park, but miss the measurements by approximately 30 MeV and 230 MeV, respectively. More importantly, the mass difference between the two states has been measured precisely [3] M[Zb (10650)] − M[Zb (10610)] ≃ 45 MeV. The expression for this mass difference using the Hamiltonian (3.2) is: r ∆ M[Zb (10650)] − M[Zb (10610)] = 2 ( + (κbq )3¯ − κbq¯ )2 + (κqq¯ − κbb¯ )2 . (3.17) 2 The smallest value for the mass difference (140 MeV) is obtained for ∆ = 0, which goes up to 247 MeV for ∆ = 202MeV. Both are larger than the measurements. Thus, the Belle data suggests that the Hamiltonian used here has to be augmented with an additional contribution. As the masses of the observed states Zb (10610) and Zb (10650) are rather close to the thresholds M(B) + M(B∗ ) and 2M(B∗ ), respectively, this suggests that the threshold effects may impact on the masses and mass differences presented here. 9


Ahmed Ali

11 320 Lb Lb

11 257 H1-- L 11 227 H1-- L 11 133 H1-- L

10 890 H1-- L

10 845

B* B * B B_* B B

10 370

10 528 H0++ L

10 504 H1++ L

0

10 520 H2++ L

10 386 H1+- L

10 385 H0++ L

++

10 527 H1+- L

++

1

1+-

1--

2++

Figure 3: Tetraquark mass spectrum with the valence quark content [bq][b¯ q] ¯ with q = u, d, assuming isospin PC −− symmetry. The value 10890 is an input for the lowest J = 1 tetraquark state Y[bq] . All masses are given in MeV. (From [25].)

4. Tetraquark-based analysis of the processes e+ e− → ϒ(1S)(π +π − , K + K − , ηπ 0 ) The cross sections and final state distributions for the processes e+ e− → ϒ(1S)(π + π − , K + K − , ηπ 0 ) near the ϒ(5S) have been presented in the tetraquark picture in [18] improving the results on the process e+ e− → ϒ(1S)π + π published earlier [17]. The distributions for the process e+ e− → ϒ(2S)π + π calculated in [17] had a computational error, which has been corrected in the meanwhile (see the Erratum in [17]). These analyses are briefly reviewed in this section. Concentrating on the processes e+ e− → ϒ(1S)(π + π − , K + K − , ηπ 0 ), there are essentially three important parts of the amplitude to be calculated consisting of the following: (i) Production mechanism of the J PC = 1−− vector tetraquarks in e+ e− annihilation. To that end, we derive the equivalent of the Van-Royen-Weiskopf formula for the leptonic decay widths of the tetraquark states Y[bu] and Y[bd] made up of a diquark and antidiquark, based on the diagram shown in Fig. 4 (left-hand frame). 24α 2 |Q[bu/bd] |2 2 (1) 2 Γ(Y[bu/bd] → e+ e− ) = κ R11 (0) . (4.1) mY4b Here, Q[bu] = 1/3 and Q[bd] = −2/3 are the electric charges of the constituent diquarks of the Y[bu] and Y[bd] , α is the fine-structure constant, the parameter κ takes into account differing sizes of the (1)

tetraquarks compared to the standard bottomonia, with κ < 1 anticipated, and |R11 (0)|2 = 2.067 GeV5 [34] is the square of the derivative of the radial wave function for χb (1P) taken at the origin. Hence, the leptonic widths of the tetraquark states are estimated as Γ(Y[bd] → e+ e− ) = 4 Γ(Y[bu] → e+ e− ) ≈ 83 κ 2 eV ,

(4.2)

which are substantially smaller than the leptonic width of the ϒ(5S) [5]. This is the reason why the states Y[bd] and Y[bu] are not easily discernible in the Rb -scan. Between the two, Y[bd] production 10


Ahmed Ali

dominates and should be searched for in dedicated experiments. However, as the decays ϒ(5S) → ϒ(nS)π + π − are Zweig-suppressed in the conventional Quarkonia descriptions, and hence have small branching ratios, the signal-to-background is much better for the discovery of the Yb (10890) in the states ϒ(nS)π + π − . These, in fact, are the discovery channels of the Yb (10890) [13]. (ii) The decay amplitudes for Yb (10890) → ϒ(1S)(π + π − , K + K − , ηπ 0 ) have non-resonant (continuum) contributions, as depicted in Fig. 4 (middle frame). They are parametrised in terms of two a priori unknown constants A and B , following [14]: 2A B 3(q0 )2 k10 k20 − |q|2 |k|2 (k1 · k2 ) + , f P f P′ f P f P′ 3s B |q|2 |k|2 , =− f P f P′ s

M01C = M02C

(4.3)

where the subscript 0 denotes the I = 0 part of the amplitudes, the superscripts 1C and 2C correspond to the S- and D-wave continuum contributions, respectively, fP(′) is the decay constant of P(′) , and |q|, k10 and k20 are the magnitude of the three momentum of Yb and the energies of P and P′ in the PP′ rest frame, respectively. Using SU(3) symmetry results in the relations involving √ the various I = 0 and I = 1 amplitudes: M01C,2C (ϒ(1S)K + K − ) = ( 3/2) M01C,2C (ϒ(1S)π + π − ), √ M11C,2C (ϒ(1S)K + K − ) = M01C,2C (ϒ(1S)K + K − ) and M11C,2C (ϒ(1S)ηπ 0 ) = 2 M11C,2C (ϒ(1S)K + K − ). We note that, in general, there is a third constant also present in the non-resonant amplitudes, characterising the term depending on the polarisation of the Yb . However, being suppressed by 1/mb , this is ignored. (iii) The resonant contributions, shown in the right-hand frame of Fig. 4, are expressed by the Breit-Wigner formula: MIR =

gRPP′ gY I ϒ(1S)R ge+ e−Y 0 b

b

2 − m2 + i m Γ MPP ′ R R R

eiϕR ,

(4.4)

where I = 0 for R = σ , f0 and f2 , and I = 1 for R = a00 . The couplings for the scalar resonances S are defined through the Lagrangian L = gSPP′ (∂µ P)(∂ µ P′ ) S + gYb ϒ(1S)S Ybµ ϒµ S, while those for µν µν the f2 are defined via L = 2g f2 PP′ (∂µ P)(∂ν P′ ) f2 + gYb ϒ(1S) f2 Ybµ ϒν f2 . The couplings gRPP′ and gY I ϒ(1S)R have mass dimensions −1 and 1, respectively. For the σ , f0 and a00 , we adopt the Flatté b model [35] and the details can be seen in [18]. With this input, a simultaneous fit to the binned ϒ(1S)π + π − data for the Mπ + π − and cos θ √ distributions measured by Belle at s = 10.87 GeV [12] were undertaken. Normalizing the diseπ + π − /d cos θ , where σ eπ + π − ≡ eπ + π − /dMππ and d σ tributions by the measured cross section: d σ Belle Belle σϒ(1S)π + π − /σϒ(1S)π + π − with σϒ(1S)π + π − = 1.61 ± 0.16 pb [12], the results are shown in Fig. 2 (histograms) and provide a good description of both the dipion mass spectrum and the angular distribution. The normalized MK+ K− and Mηπ 0 distributions are shown in Fig. 5 (a) and Fig. 5 (b), respectively. In these figures, the dotted (solid) curves show the dimeson invariant mass spectra from the resonant (total) contribution. Since these spectra are dominated by the scalars f0 + a00 and a00 , respectively, there is a strong correlation between the two cross sections. This is shown eηπ 0 are plotted resulting from eK+ K− and σ in Fig. 5 (c), where the normalized cross sections σ 2 the fits (dotted points) which all satisfy χ /d.o.f. < 1.6 [18]. The current Belle measurement 11


Ahmed Ali

Υ(nS)

Yb

e+

q

P

e−

Υ(nS)

Yb

e+

e−

R

P′

P P′

40

(a)

1.5

1

0.5 0

1.1

1.2

1.3

MK +K − [GeV]

1.4

(c)

6

20 10

2

0 1

8

(b)

30

f 0 σ ηπ

2

f 0 /dM 0 dσ ηπ ηπ

f + − /dM + − dσ K K K K

Figure 4: Left frame: Van Royen-Weiskopf Diagram for the production of a J PC = 1−− tetraquark Yb with the quark content [bu][b¯ u] ¯ in the process e+ e− → γ ∗ → Yb . Middle frame: Continuum contribution in the process e+ e− → Yb → ϒ(nS)PP′ . Right frame: Resonance contribution in the process e+ e− → Yb → ϒ(nS)PP′ . (Figures based on [18].)

4

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Mηπ0 [GeV]

0

0

0.1

0.2

0.3

0.4

f + − σ K K

Figure 5: Predictions (a) of the MK + K − distribution for e+ e− → Yb → ϒ(1S)K + K − , (b) of the Mηπ 0 distribution for e+ e− → Yb → ϒ(1S)ηπ 0 and (c) of the correlation between the cross sections of ϒ(1S)K + K − and ϒ(1S)ηπ 0 , normalized by the measured cross section for the ϒ(1S)π +π − mode. In (a) and (b), the dotted (solid) curves show the dimeson invariant mass spectra from the resonant (total) contribution. In (c), the red dots represent predictions from the fit solutions satisfying χ 2 /d.o.f. < 1.6. The shaded (green) band shows the current Belle measurement σeK + K − = 0.11+0.04 −0.03 [12]. (From [18].) +0.04 σeK+ K− = 0.11−0.03 [12] is shown as a shaded (green) band on this figure. The tetraquark model [18] is in agreement with the Belle measurement, and prediction 1.0 . σeηπ 0 . 2.0. will be further tested eηπ 0 is measured. Another important test of the tetraquark model as and when the cross section σ is [12]

Q2[bu] σϒ(1S)K+ K− 1 = 2 = . σϒ(1S)K0 K¯ 0 Q[bd] 4

(4.5)

This remains to be tested. Finally, the corrected analysis [17] of the dipion invariant mass spectrum and the helicity angle distribution (in cos θ ) for the process Yb (10890) → ϒ(2S)π + π − are shown in Fig. 6, in which the normalization is given by the measured partial decay width Γ[Yb (10890) → ϒ(2S)π + π − ] = 0.85 ± 0.7 ± 0.16 MeV [13]. The dipion invariant mass spectrum is well accounted for also in this process (χ 2 /d.o.f. = 12.6/7), but not the the angular distribution dΓ/d cos θ . These distributions are being reevaluated taking into account the resonances Zb (10610) and Zb (10650). As a tentative summary of the tetraquark interpretation of the Belle data on e+ e− → (ϒ(nS)π + π − and e+ e− → hb (mP)π + π − is that the existing analysis are encouraging and there exists a prima facie case of its validity. However, the missing contributions from the charged tetraquarks in the analysis of the e+ e− → (ϒ(nS)π + π − data have to be incorporated and the fits of the e+ e− → hb (mP)π + π − data have to be undertaken to get a definitive answer. I would like to thank Robert Fleischer and the organisers of the Beauty 2011 conference for a very exciting meeting in Amsterdam. I also thank Christian Hambrock, Satoshi Mishima and Wei Wang for their help in preparing this talk and helpful discussions. 12


Ahmed Ali

--

---

˜ dΓ/dcosθ/Γ(2S)

˜ [GeV−1] dΓ/dmπ+π− /Γ(2S)

1.0 4

3 --

--

--

-2 -1

-0.6

---

---

0.4

--

--

--

0.8

--

0.2

--

--

--

-0 0.2

-0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.0 -1.0

-0.5

0.0

mπ + π − [GeV]

0.5

1.0

cos θ

Figure 6: Dipion invariant mass (mππ ) distribution (left-handed frame) and the cos θ distribution (righthanded frame) measured by the Belle collaboration for the final state ϒ(2S)π + π − [12] and the corresponding theoretical distributions (histograms) based on the tetraquark interpretation of the YB (10890). (From [17].)

References [1] S. L. Olsen, Nucl. Phys. A 827, 53C (2009) [arXiv:0901.2371 [hep-ex]]; A. Zupanc [for the Belle Collaboration], arXiv:0910.3404 [hep-ex]. [2] N. Brambilla et al., Eur. Phys. J. C 71, 1534 (2011). [3] I. Adachi et al. [Belle Collaboration], arXiv:1105.4583 [hep-ex]. [4] I. Adachi et al. [Belle Collaboration], arXiv:1103.3419 [hep-ex]. [5] C. Amsler et al. [Particle Data Group], Phys. Lett. B 667, 1 (2008). [6] S. Godfrey, J. L. Rosner, Phys. Rev. D66, 014012 (2002). [arXiv:hep-ph/0205255 [hep-ph]]. [7] J. P. Lees et al. [BABAR Collaboration], arXiv:1102.4565 [hep-ex]. [8] M. B. Voloshin, Sov. J. Nucl. Phys. 43, 1011 (1986). [9] S. Godfrey, J. Phys. Conf. Ser. 9, 123-126 (2005). [hep-ph/0501083]. [10] T. K. Pedlar et al. [CLEO Collaboration], Phys. Rev. Lett. 107, 041803 (2011). [11] S. Dobbs et al. [ CLEO Collaboration ], Phys. Rev. Lett. 101, 182003 (2008). [arXiv:0805.4599 [hep-ex]]. [12] K. F. Chen et al. [Belle Collaboration], Phys. Rev. Lett. 100, 112001 (2008). [13] I. Adachi et al. [Belle Collaboration], Phys. Rev. D82:091106(R) (2010). [14] L. S. Brown and R. N. Cahn, Phys. Rev. Lett. 35, 1 (1975); M. B. Voloshin, JETP Lett. 21, 347 (1975) [Pisma Zh. Eksp. Teor. Fiz. 21, 733 (1975)]; V. A. Novikov and M. A. Shifman, Z. Phys. C 8, 43 (1981); Y. P. Kuang and T. M. Yan, Phys. Rev. D 24, 2874 (1981). [15] K. Gottfried, Phys. Rev. Lett. 40, 598 (1978). [16] A. Sokolov et al. [Belle Collaboration], Phys. Rev. D 79, 051103 (2009). [17] A. Ali, C. Hambrock and M. J. Aslam, Phys. Rev. Lett. 104, 162001 (2010); 107, 049903 (E) (2011). [18] A. Ali, C. Hambrock and S. Mishima, Phys. Rev. Lett. 106, 092002 (2011). [19] A. E. Bondar, A. Garmash, A. I. Milstein, R. Mizuk and M. B. Voloshin, arXiv:1105.4473 [hep-ph]. [20] M. B. Voloshin, [arXiv:1105.5829 [hep-ph]]. [21] M. Cleven, F. -K. Guo, C. Hanhart, U. -G. Meissner, [arXiv:1107.0254 [hep-ph]].

13


Ahmed Ali

[22] Y. Yang, J. Ping, C. Deng, H. -S. Zong, [arXiv:1105.5935 [hep-ph]]. [23] Z. -F. Sun, J. He, X. Liu, Z. -G. Luo, S. -L. Zhu, [arXiv:1106.2968 [hep-ph]]. [24] S. -K. Choi, S. L. Olsen, K. Trabelsi, [arXiv:1107.0163 [hep-ex]]. [25] A. Ali, C. Hambrock, I. Ahmed and M. J. Aslam, Phys. Lett. B 684, 28 (2010). [26] N. V. Drenska, R. Faccini and A. D. Polosa, Phys. Lett. B 669, 160 (2008); N. V. Drenska, R. Faccini and A. D. Polosa, Phys. Rev. D 79, 077502 (2009). [27] R. L. Jaffe, Phys. Rept. 409, 1 (2005) [Nucl. Phys. Proc. Suppl. 142, 343 (2005)]. [28] L. Maiani, F. Piccinini, A. D. Polosa and V. Riquer, Phys. Rev. D 71, 014028 (2005) [arXiv:hep-ph/0412098]. [29] A. De Rujula, H. Georgi and S. L. Glashow, Phys. Rev. D 12, 147 (1975). [30] R. L. Jaffe and F. Wilczek, Phys. Rev. Lett. 91, 232003 (2003). [31] N. Drenska, R. Faccini, F. Piccinini, A. Polosa, F. Renga and C. Sabelli, Riv. Nuovo Cim. 033, 633 (2010). [32] D. Ebert, R. N. Faustov and V. O. Galkin, Mod. Phys. Lett. A 24, 567 (2009) [arXiv:0812.3477 [hep-ph]]. [33] Z. G. Wang, Eur. Phys. J. C 67, 411 (2010) [arXiv:0908.1266 [hep-ph]]. [34] E. J. Eichten and C. Quigg, Phys. Rev. D 52, 1726 (1995). [35] S. M. Flatte, Phys. Lett. B 63, 224 (1976).

14